Question

In Exercise 9 of Section 12.2 you proved that \(f: \mathbb{R}-\\{2\\} \rightarrow \mathbb{R}-\\{5\\}\) defined by \(f(x)=\frac{5 x+1}{x-2}\) is bijective. Now find its inverse.

Short answer

The inverse function of \(f(x) = \frac{5x + 1}{x - 2}\) is \(f^{-1}(x) = \frac{1 + 2x}{x - 5}\).

Step-by-step solution

Identify the Function

The function given is \(f(x) = \frac{5x + 1}{x - 2}\). We need to find its inverse. An inverse function undoes the operation of the original function.

Swap \(x\), \(y\)

The first step in finding the inverse of a function is to replace the function \(f(x)\) with \(y\), which gives us \(y = \frac{5x + 1}{x - 2}\). Now swap the roles of \(x\) and \(y\), which results in \(x = \frac{5y + 1}{y - 2}\).

Solve for \(y\)

We rearrange this equation to solve for \(y\). To do this, first cross-multiply to get rid of the fraction: \(x(y - 2) = 5y + 1\). Expanding the left-hand side results in \(xy - 2x = 5y + 1\). Now, bring terms with \(y\) on one side: \(xy - 5y = 1 + 2x\). Factor out \(y\): \(y(x - 5) = 1 + 2x\). Solving for \(y\), we get \(y = \frac{1 + 2x}{x - 5}\).

Write the Inverse Function

Replace \(y\) with \(f^{-1}(x)\) to write the inverse function. Therefore, the inverse function is \(f^{-1}(x) = \frac{1 + 2x}{x - 5}\). This is the function that will reverse the operation of the initial function.